Did you read the Wikipedia article? If so, please be more specific about what parts you need help with. There doesn't seem to be much point in somebody rewriting that article as an answer.
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David RicherbyMar 7 '14 at 22:25

Wing generates lift by creating area of higher pressure below and area of lower pressure above.

At the wing tip, some air flows around the tip, reducing the pressure differential and thus the efficiency of the wing. This is called transverse flow. The wing tip reduces this flow, thus improving the efficiency.

The effect of wing tip is actually similar to making the wing a bit longer, but since the wing tip does not produce lift itself, it is less stressed and therefore can be lighter, even though the extension would be about 1/3 shorter for comparable efficiency.

Winglets, raked wingtips, fences, sharklets all do the same thing; reduce drag. When the wing is working hard (high AoA), it will produce a lot of this kind of drag since vortecies of all kinds will be stronger. Winglets makes the passage of air from the bottom of the wing to the lower pressure top of the wing more difficult, thus reducing induced drag.

The big manufacturers explored offering winglet retrofit kits to their customers but at the time, it wasn't economical since fuel was so cheap. These days, however, the retrofits pay for themselves fairly quickly even on airplanes wings that have a fairly high aspect ratio. Every 737/757/767 guy that I talk to says they get in the neighbor hood of a 4-6% boost in fuel savings.

On an tangentially related topic, the hump from the satellite internet antennae has a negligible impact on performance, less than 1% according to the few guys that I talked to.